What are the Controls on Distributions of Local Wave Activity?

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What are the Controls on Distributions of Local Wave Activity?

Claire Valva

A thesis submitted in partial fulfillment for departmental honors in geophysical sciences

Faculty readers: Professor Noboru Nakamura and Professor Susan Kidwell

Department of Geophysical Sciences, The University of Chicago

2020

Abstract

The meandering of the jet stream is a main cause of weather variation in the midlatitudes. Very large and persistent meandering of the jet stream can cause weather extremes. One way to measure the meandering of the jet stream is through finite-amplitude local wave activity (LWA), a measure of the displacement of quasigeostrophic potential vorticity (Huang and Nakamura 2016). LWA tends to correlate negatively with the zonal wind and its large value is particularly useful at identifying and blocking events.

In this study, we examine the observed probability distribution of LWA in the midlatitudes of both hemispheres, as well as demonstrate the correlation between weather extremes and high LWA. We in-vestigate what controls LWA’s seasonal and interhemispheric variability. Further, using both the one-dimensional ’traffic jam’ model of Nakamura and Huang (2018) and the quasigeostrophic two-layer model, we explore sensitivity of LWA distributions to variation of the stationary wave amplitude, transient eddy forcing, and the jet speed, among other parameters.

1 Introduction

Eastward winds in the midlatitudes form a jet stream in the middle to upper troposphere. The jet stream meanders over thousands of kilometers, a meandering which is caused by weather systems (cyclones and anticylones). As the jet stream is a dynamic boundary between the cold subpolar airmass and the warm subtropical airmass, its undulation brings about periodic fluctuations in air temperature when observed at a fixed location. Particularly persistent meandering of the jet stream can disrupt the passage of transient waves: an occurrence which is known as blocking. A block can last for a few days to over a week, and is associated with anomalous and extreme weather events in the midlatitudes. For example, the 2003 European Heat Wave (figure 1) — which caused tens of thousands of deaths — was at least partially due to strong anticyclonic blocking [1].

Figure 1: Demonstration of link between high values of LWA and extreme weather: (left) histograms of the de-viation from JJA climatolog-ical mean LWA, geopoten-tial height (GPH), and sur-face temperatures during July and August of 2003 — the time period over which the deadly European Heat Wave took place — between 45◦and 48◦N and 0◦to 4.5◦E.(right) Mean temperature difference from seasonal climatology. Carl-Gustaf Rossby, who founded the Department of Meteorology at the University of Chicago, was the first to characterize the jet stream’s meandering as a kinematic wave in the vorticity field [12]. Vorticity, or the spin rate of air parcels about the vertical axis, of the atmosphere increases with latitude because the local vertical will align more closely with the rotational axis of the planet as a parcel moves closer to the pole. Meridional displacement of vorticity induces a flow that sustains the displacement in a waveform: called the Rossby wave.

Currently there is incomplete mechanistic understanding of blocking events, and the definition of blocking is still somewhat subjective. Theories for the causes of the onset of persistent blocking anomalies include resonance between stationary waves and forcing as well as diffluent flow [3], but ultimately these form an incomplete theoretical understanding. (Reasons include difficulty in direct verification with data or lack of predictive skill.) Geopotential height anomaly (difference of geopotential height from the zonal mean) is a traditional metric for measuring the meandering of the jet stream. However, geopotential height does not obey a simple governing equation, nor does the deviation of geopotential height from zonal mean explain the causes for the meandering. A more fundamental quantity to large scale dynamics is quasi-geostrophic potential vorticity (QGPV) — essentially the sum of the relative vorticity and the vorticity from the rotation of the planet (see 1.1.1) — which obeys a nice conservation equation.

It is then natural to want to measure the meandering of the jet stream with QGPV, which brings us to finite-amplitude local wave activity (LWA); LWA is the amplitude of Rossby waves measured by the meridional displacement of quasigeostrophic potential vorticity (PV) [6]. As we know the budget for LWA, LWA can be considered simpler metric to understand atmospheric blocking.

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If LWA is an effective way to measure the meandering of the jet stream, extreme highs in LWA will be linked to extreme weather events. For example, a 95th percentile deviation from mean seasonal temperature and geopotential height are twice as likely to occur on days where LWA is also of the 95th percentile of magnitude. It is then important to understand the distribution of LWA values in climatological mean, as well as how the distribution changes under different conditions. In this work we will seek to understand LWA distributions through observations and several modeling experiments with the following goals:

i To demonstrate the correlation between weather extremes and high LWA as well as understand the distribution of LWA as well as related statistics of the meandering of the jet stream.

ii To understand the factors which affect the probability distributions of LWA using two models of different complexity.

1.1 Local Wave Activity Formalism

For completeness sake, I will describe the technical aspects of LWA below. Readers who are interested in only the application or results of this study may skip the following section, and look over only the following paragraph:

For each meridional coordinate (y), one can define an expected value of potential vorticity which is constant on the zonal circle. The actual isoline of that value of potential vorticity will meander over this line, and the area between the actual isoline and the expected meridional coordinate corresponds to the amplitude of the meandering over this line, the quantity of meandering is LWA. In this way, LWA quantifies the amplitude of the waviness of the Rossby waves in the atmosphere. If we integrate over the entire meridian, we retrieve finite amplitude wave activity (FAWA) which measures the waviness of an isoline of potential vorticity as coordinatey. (For a more visual representation of the physical derivation of LWA and FAWA, see Figure 2.)

In [8], FAWA was formalized, where FAWA quantifies waviness in the PV contours and the associated modification in zonal mean-flow. However, FAWA does not distinguish the longitudinal location of an isolated large-amplitude event such as blocking. LWA is the generalization of FAWA to be a function of longitude as well. The derivation of LWA from FAWA is shown in Figure 2.

Let Qbe a contour of constant potential vorticity, the function Q(y, z) is the potential vorticity equiv-alent latitude relation which could be considered to define a zonally symmetric, time-invariant “reference state.”Potential Vorticity (PV), a conserved quantity, is essentially the sum of relative vorticity, the spin rate relative to the rotating planet, and planetary vorticity, from the spin rate of the earth. PV is written as

q(x, y, z, t) =ζ+f(1 +ez/H ∂ ∂z{e

−z/Hθ−θ(z)˜

dθ/dz˜ } (1.1.1)

where f(y) is the Coriolis parameter, ζ is relative vorticity, θ is potential temperature where ˜θ(z) is the global horizontal average, andH is the constant scale height.

The potential vorticity contourQis displaced locally (x, y, z) to (x, y+η(x, y, z, t), z) whereη(x, y, z, t) is defined to be positive in the northwards direction, with latitudey, pressure pseudoheightx, and timet. We can then define LWA as ˜A∗_{(x, y, z, t) as}

˜

A∗(x, y, z, t)≡ −

Z η(x,y,z,t)

0

qe(x, y+y0, z, t)dy0, (1.1.2)

whereqe is the eddy field defined between (x, y, z) and (x, y+η, z) asqe(x, y+y0, z, t)≡q(x, y+y0, z, t)−

Q(y, z). We then can write the equation for FAWA as:

A∗(y, z, t) =− 1 Lx Z Lx 0 Z η 0

[qe(x, y+y0, z, t)+Q(y, z)]dy0dx=

1 Lx Z Lx 0 [ ˜A∗(x, y, z, t)−ηQ(y, z)]dx= ˜A∗_{(x, y, z, t),} (1.1.3)

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where h·i denotes zonal averaging and Lx is the length of the zonal circle.) In essence, LWA quantifies

longitude-by-longitude contributions of FAWA and will recover FAWA upon zonal averaging.

We also briefly note that LWA can be broken into two quantities: cyclonic LWA, which is the cyclonic wave activity residing to the south of the band of equivalent latitude, and anticylonic LWA which is the wave activity that resides to the north of the band of equivalent latitude.

1.2 LWA Budget

As mentioned, one of the advantages of LWA is that LWA obeys a more explicit governing equation (the momentum equation for a Rossby wave packet) than geopotential height, which requires inversion of potential vorticity??. The column budget of LWA reads:

∂ ∂thAicosϕ=− 1 acosϕ ∂hFλi ∂λ | {z } I + 1 acosϕ ∂ ∂ϕ0huevecos 2_(ϕ₊_ϕ0_)i | {z } II +fcosϕ H ( veθe ∂θ/∂z˜ )z=0 | {z } III +hAi˙ cosϕ | {z } IV (1.2.1)

Figure 2: A schematic diagram (on the x–y plane) of the surface integral domainsD1andD2to derive FAWA

[8]. The horizontal dashed lines indicate the equivalent latitude corresponding to the PV contour shown such that the pink and blue areas are the same. A schematic diagram illustrating how to compute the local finite-amplitude wave activity in 1.1.2 and 1.1.3. The wavy curve indicates a contour of PV, above which the PV values are greater than below. Inside the red lobesqe≥0, and inside the blue lobes qe≤0. (The red lobes

correspond to cyclonic LWA and the blue to anticyclonic LWA.) Four points are chosen to illustrate how the domain of integral is chosen: ˜A∗(x1, y) =−R_W

1+qe(x, y+y 0_{, z, t)dy}0_{; ˜}_A∗_(x 2, y) =−R_W 2−qe(x, y+y 0_{, z, t)dy}0_: ˜ A∗(x3, y) = R W3−qe(x, y+y 0_{, z, t)dy}0₋R W3+qe(x, y+y 0_{, z, t)dy}0_{; ˜}_A∗_(x 2, y) = R W4−qe(x, y+y 0_{, z, t)dy}0 _(Both

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Where the formhFλiis as follows:

hFλi=huREFAcosϕi

| {z } FI −ah Z ∆ϕ 0 ueqecos(ϕ+ϕ0)dϕ0i | {z } FII +cosϕ 2 hv 2 e−u 2 e− R H eκz/H ∂θ/∂z˜ θ 2 ei | {z } FIII (1.2.2)

The explicit representation of nonconservative dynamics (such as local diffusive flux of potential vorticity) are not present in the LWA budget. However, if observed data does significantly deviate with this theoretical budget, we can then interpret this as evidence of significant nonconservative processes present in the dynamics of those observations.

2 Observations

2.1 Climatology

Figure 3 summarizes seasonal climatology of column averages LWAhAicosϕ. Generally mean LWA values in the Southern Hemisphere are lower than in the Northern Hemisphere. (We note that seasonal extremes match this trend, where the 95th percentile values of LWA are generally higher in the Northern Hemisphere than the Southern Hemisphere.) In general, anticyclonic wave activity dominates — although the difference between average magnitudes of anticylonic and cyclonic LWA is higher in the Northern Hemisphere. However, in several locations cyclonic wave activity is predominant such as off the coast of Japan and the Southern Atlantic - Indian Ocean sector of the austral storm track. Local maxima in the Northern Hemisphere include the western coast of North America as well as western Europe. In the Southern Hemisphere, local highs of LWA include eastern South America and just north of Antarctica.

In the Northern Hemisphere winter, maxima in LWA are found in the Euro-Atlantic sector of the storm track, over the Bering Sea, and the Eastern Pacific. In the Summer, maxima are found in the midlatitude Pacific, and the Euro-Atlantic section of the storm track. In the Southern Hemisphere winter, LWA maxima are found over the Antarctic Circumpolar current, and over landmasses. In Summer, maxima are found in the Pacific midlatitudes.

In the Northern Hemisphere, values of LWA are highest in the Summer (June/July/August) and Winter (December/January/February), as in the Southern Hemisphere. However, Northern Hemisphere winter (DJF) has higher values of LWA than in summer (JJA), while in the Southern Hemisphere the increases in local wave activity during winter and summer are more balanced (in a sense). In the Southern Hemisphere summer (DJF), the local maximum of LWA tends more south than in the winter (JJA).

Climatologies of zonal wind speed are plotted in figure 4. It is well known that the transient part of zonal wind is negatively correlated with LWA. This inverse correlation is a results from the fact that the sum of sum of phase-averaged barotopic LWA, surface LWA, and zonal wind is a constant sum in the conservative limit (as found in equation 36 [6]). As such, increases in wave amplitude occurs at the expense of local zonal wind. However, when comparing the wind speed climatology with the climatology of LWA, we do not see this inverse relationship everywhere. It is somewhat pronounced — when searched for — in some areas and seasons. For example, in Northern Hemisphere winter, there is a zonal wind maximum in the Western Pacific, which corresponds to a local minimum in LWA.

In figure 5, we plot the climatologies of surface temperatures and geopotential height (at 300 hPa). In the Northern Hemisphere, geopotential height shows the most zonal variance in the winter (and the least in the summer), where geopotential height is at a maximum by the west coast of North America and in the western Atlantic and western Europe (into West Asia). The corresponding lows in geopotential height are present in eastern North America and East Asia. Similarly, positive temperature anomalies over western North America are most distinct in the winter and the summer. A distinct zonal temperature maximum over the eastern Atlantic is present in DJF, but not present in the summer. Instead in JJA, there is a temperature maximum in eastern Europe and parts of northern Asia, where there are relative lows in the winter.

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Figure 3: Seasonal climatology of LWA, based on 40 years of LWA observations (1979-2018, derived from ERA-Interim Reanalysis).

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Figure 4: Seasonal climatology of zonal velocity based on 40 years (1979-2018) of observations (ERA-Interim). It is known that zonal velocity is inversely correlated to LWA.

In the Southern Hemisphere, relative geopotential highs are mostly consistent between seasons, as are variations from zonal temperatures. Continental temperatures are generally higher than ocean temperatures in the Southern Hemisphere, except for over Antarctica where there is a large amount of continental variation in temperature. (This is to be mostly expected, as there are no oceans in that southern latitude, and thus the regularization of temperature expected from oceans will not affect temperatures zonally.) Further, besides the modulation of temperature by land surfaces, the overall patter of surface temperatures is positively correlated with geopotential height, as could be expected from the hypsometric relation (which, in essence, relates the equivalent thickness of an atmospheric layer to temperature).

As we claim that LWA is related to atmospheric blocking, and then from there to weather variables, we can then ask if any obvious correspondences exist between climatologies in LWA (figure 3) and these weather variables (figure 5). In the Northern Hemisphere, relatively high zonal temperatures correspond to relative maximums in total LWA in both the summer and winter. In particular, the relatively high values of LWA in summer correspond to the relative temperature highs in eastern Europe and parts of northern Asia, while the relative LWA high in winter corresponds with the relatively high temperatures in the Atlantic. In the Southern Hemisphere, surface temperature highs in climatology over landmasses, correspond loosely to the highs in anticyclonic LWA. Relative maxima in the southern Pacific in geopotential height also correspond to similar highs in both cyclonic and anticyclonic LWA.

2.2 LWA Distributions

While seasonal climatology depicts a slowly varying state of the atmosphere in response to external forcing, blocking and other weather anomalies arise from the internal variability of the atmosphere on intraseasonal timescales. To capture the statistics of weather variables more fully, it is useful to examine probability distribution of variables. Characteristic probability distributions of midlatitude JJA LWA based on daily averages of 6-hourly data are shown in figure 6, both for the entire zonal circle as well as for a segment between 230◦to 320◦E (which is roughly over North America). Although we only plot one season and hemisphere, for a relatively small meridional segment, the following discussion of the shapes of the probability distributions of LWA hold for both hemispheres, as well as all seasons (if both median and LWA extreme values vary).

Distributions of LWA are smooth, unimodal, with a evident right skew. The skew of total LWA is less than the skew of either cyclonic LWA or anticyclonic LWA. Anticyclonic LWA has a generally larger median as well as larger values for the 95th percentile LWA than cyclonic LWA; given the relative climatology

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Figure 5: Seasonal climatologies for Surface Temperatures and Geopotential Height at 300 hPa. Climatology is presented as the difference from zonal means because temperature and geopotential height vary strongly with latitude to better represent local variation. Climatologies are based on ERA-Interim reanalysis product from 1979-2018.

of cyclonic and anticyclonic LWA as noted in section 2.1, where anticyclonic LWA generally greater than cyclonic LWA, this result is expected. We make the comparison of LWA distributions over the entire zonal circle and a segment not necessarily for a quantitative judgement of exactly how the distribution shifts zonally — but rather to display a consistency between the shapes of LWA probability distributions in the majority of locations.

Notably, all of these distributions can be well fit to a gamma distribution. A gamma distribution — a general type of statistical distributions — arises naturally in processes where the events are Poisson distributed [5]. As a result, we hypothesize that the injection of LWA into the jet could be modeled as a poisson process.

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Figure 6: Histograms of Northern Hemisphere mid-latitude LWA in JJA from observations from 1979 to 2018. Values of LWA are from 40◦ _{to 50}◦ _{N. Left column}

shows histograms over entire zonal circle, and right column plots observations from 230◦ to 320◦ E. The y-axis is in units of “density” or percent-age of values which achieve an LWA value.

2.3 Covariance of LWA and Weather Variables

Generally, background jet velocity (denoted UB in the accompanying figures) varies negatively with high values of LWA. We also investigate the relation of extreme (high) values of LWA and extreme (low) values of background velocity to weather variables.

In Figure 7, we plot the correlation of LWA —both cyclonic, anti-cyclonic, and total — with two weather variables: surface temperatures and geopotential height (at 300 hPa). The correlation is the coefficient r from a linear regression between 40 years of data (1979 - 2018). The linear regression was performed post-detrending to remove interannual trends in variables (such as average increasing temperatures over the time period).

In the Northern Hemisphere, LWA is relatively strongly correlated with geopotential height as well as surface temperatures. Generally, in the Northern Hemisphere geopotential height increases with anticyclonic LWA and decreases with cyclonic LWA; as the magnitude of anticyclonic LWA is higher on average than than cyclonic LWA, this results in geopotential height increasing with LWA in most locations. Similarly, in the Southern Hemisphere, we find that both surface temperatures and geopotential height increase with anticyclonic LWA, and decrease with cyclonic LWA. However, as the magnitude of anticyclonic and cyclonic LWA are more similar in the Southern Hemisphere, geopotential height increases with LWA over landmasses, but not necessarily over the Antarctic Circumpolar Current. We find similar correlations in the relation between LWA and surface temperatures in both the Northern and Southern Hemispheres (if to a slightly lesser degree in the Northern Hemisphere); in the Northern Hemisphere surface temperatures increase with LWA

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Figure 7: Correlation between June/July/August (JJA) and Decem-ber/January/February (DJF) geopoten-tial height (at 300 hPa) and surface temper-atures and local wave activity.

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in most locations — particularly in JJA), but the correlation between total LWA and surface temperatures are more variable in the Southern Hemisphere.

Figure 8: (A) Plotted is mean amplitude of the spectra of the residual forcing term in the LWA budget from midlatitude Northern Hemisphere JJA observations, using 40 years of ERA-Interim reanalysis (1979-2018). This spectra was used to prescribe the transient eddy forcing in the one-dimensional traffic model: thirty wavenumber spectra pairs (with corresponding amplitude) are chosen randomly (with twice the chance of choosing a wavenumber less than 3 and a frequency less than 1day−1_{) for the transient forcing of the model,}

where phase is randomly chosen in both time and space (x). Note that low frequencies and wavenumbers are dominant, and the propagation of waves in the negative ”x-direction” denotes the westward propagation of waves. (B) A Hovm¨oller diagram from model output with standard parameters (U = 50ms−1_, _γ_{= 2.0,}

α= 0.55) and time given in days. This diagram begins after the model ”spin-up.”

This result brings out the dominant role of continents in the Northern Hemisphere. Due to the relatively small heat capacity of land as compared to oceans, the land surfaces are hot in the summer (and cold in the winter), which impacts the stationary wave components of LWA (as well as surface temperatures). The correlation between surface temperatures and LWA is strongest over landmasses: in particular, for cyclonic LWA in the winter and anticyclonic LWA in the summer. A likely reason for more variable correlation between surface temperatures and LWA in the Southern Hemisphere is the ocean surface damping temperature variations through heat exchange with the atmosphere.

3 Traffic Flow Model

3.1 Description

As a first-order dynamical model of atmospheric blocking, Nakamura and Huang introduced a one-dimensional nonlinear partial differential equation based on the observed budget of LWA [7]. The nonlinear partial dif-ferential equation (PDE, eq. 3.1.1) is derived from the more general conservation of LWA at a particular latitude, assuming that the atmosphere is barotropic [6].

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We can partition the amplitude of LWA asA(x, t)≡A0(x) +A(x, t), whereˆ A0(x) is the stationary wave

component of LWA and ˆAis the transient wave component. Then, equation 3.1.1 governs the evolution of the transient wave component.

∂ ∂t ˆ A(x, t) =−∂ ∂x[(C(x)−α ˆ A) ˆA] + ˆS−Aˆ τ +D ∂2_Aˆ

∂x2 wherexis longitude andt is time. (3.1.1)

where

F := (C(x)−αA) ˆˆ A (3.1.2)

C(x) :=uREF+cg−2αA0(x) (3.1.3)

The quantityF is the nonlinear zonal flux of ˆA, whereC(x) is defined to be the background group velocity (uREF +cg) modulated by stationary waves A0 from the reference state. The nonlinear zonal flux in

amplitude corresponds from the empirical relation between the advecting zonal winduand LWA:u(x, t) = UREF −αA(x, t), whereUREF is the constant zonal wind of the wave-free reference state. The parameter

αmeasures the strength of wave-zonal flow interaction, which was empirically determined to beα≈0.55 in [7]. We note that we can considerαto be the strength of nonlinearity. The length of the domain is chosen to be 28,000 km with periodic boundary conditions.

In analogy to traffic flow, C(x) plays the role of the speed limit, so if Â continues to grow F(x, t) = [C(x)−αA] ˆÂwill eventually stop growing. As Â grows past this critical value (C(x)_2α ), which we also refer to as the wave-breaking threshold,F begins to decrease, causing a runaway accumulation of wave activity if there is a continued supply of wave activity from upstream; the resulting rapid increase of Â and drop ofF characterizes block formation [7], [9].

The other three terms of 3.1.1 are ˆS(x, t) which is transient eddy forcing (this includes diabatic heating and meridional divergence of eddy momentum flux); Aˆ

τ which is the linear damping of LWA; andD ∂2Aˆ ∂x2, a

diffusion term. We derive the transient eddy forcing from the average observed spectra of midlatitude LWA in the Northern Hemisphere, see figure 8 for a more explicit representation.

While equation 3.1.1 is highly idealized, and as such does not give accurate predictions of blocks in real weather, it does represent the canonical dynamics that produce persistent anomalies in the jet stream and reproduces features of blocking when the LWA amplitude/flux threshold is reached.

Part of the utility of this model is the economy, as it is suitable for large ensemble runs in parameter sweep experiments. In the following experiment we vary 3 parameters: the amplitude of transient eddy forcing γ (where |S| ∝ˆ γ), α (the strength of nonlinearity or wave-zonal flow interaction), andUj which

is a constant jet speed, that denotes the average group velocity of Rossby waves in the wave-free reference state, which includes advection by the zonal windUREF. We defineUj≡UREF+cg. With nonconservative

processes,UREF would not be invariant in time, instead varies slowly, however this experiment is meant to

understand blocking in a canonical reference state.

We will consider α ∈ [0.15,3], γ ∈ [0.5,4], and Uj ∈ [40m/s,70m/s], where each parameter is varied

independently with the other two taking a default value. (The default values are α = 0.55, γ = 2.0, and uj= 60m/s.) We run the model for 270 days without transient eddy forcing or noise from stationary waves

to ready a steady state, and then turn on the transient forcing and noise. We repeat this experiment 200 times for each parameter combination where the parameters for phase of forcing are randomized each time. A snapshot of LWA is saved once every 6 hours. However, all results shown in this section are for a four-day average because of our interest in blocking events, where both large and persistent LWA values are important.

3.2 Results

3.2.1 LWA Extremes and the Wave-breaking Threshold

The results of this experiment are presented in figure 9 and compute the wave breaking threshold computed for parameter combination (the wave breaking threshold depends onUj andα, but notγ).

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Figure 9: Results from 1 layer-PDE model: The threshold values are denoted by along the yellow line and the 90th and 95th percentiles designate the mean between model runs of that extreme percentile value. The error bars represent the standard deviation of extremes between runs. (A) Variation of extreme LWA values as parameterαincreases (γ= 3, Uj= 60m/s); (B) Variation of extreme LWA values as parameterγ

(amplitude of transient waves) varies (α= 0.55, Uj= 60m/s); (C) Variation of extreme LWA values as the

jet speed (Uj) increases (α= 0.55,γ= 3).

Figure 9A depicts extreme (90th and 95th percentiles of LWA post spin-up period) values of LWA increasing asαdoes, with an inflection in the curve at approximatelyα≈0.55. Asαincreases, the variance of extreme values in between runs as well as the difference in the percentile values increases. The extreme values of model runs reach the wave breaking threshold only for a value of alpha greater than 0.55, indicating that the magnitude of the nonlinear term is a significant factor in LWA surpassing threshold values. As such, these results indicate that nonlinearity is a significant element of blocking behavior in the midlatitudes.

Figure 9B depicts extreme (90th and 95th percentiles of LWA post spin-up period) values of LWA, which increase withγ. As the wave breaking threshold is not dependent onγand the initial extreme values of LWA surpassed this value, all tested values of gamma resulted in wave breaking events. Similar to the behavior of the model when αincreases, asγ increases, variance of the extreme values and the difference in percentile values increased as well.

Figure 9C shows extreme values of LWA plotted against background wind speed. As background wind speed increases, LWA decreases, mirroring the negative relationship between wind speed and LWA as seen in observations. In contrast, increasing windspeed corresponds with a linear increase in the wave breaking threshold, which results in experiments run withUj<65m/shaving few to no wave breaking events. 3.2.2 Distributions

Probability distributions from the one-dimensional LWA model were similar to those found in observations, having a strong right skew that increased as values of αincreased. In fact, nonlinear forcing in the model is the key to an asymmetrical distribution of LWA values. In figure 10, we show three distributions from the model with increasing values of α. As α increases, the assymetry of the distribution becomes more pronounced. Consider a analogous, but linear, version of this PDE:An+1=An−A_τn +Fn whereAn is the

amplitude of LWA at time n, τ is a linear damping parameter, and Fn is a Poisson process which injects

LWA. If LWA behaved under these linear dynamics, the distribution is a symmetrical Gaussian as long as the damping timescale is sufficiently longer than the averaging timescale. When we further consider that the nonlinear parameterαmust pass a minimum value for values of LWA to pass the wave-breaking threshold, and as such induce blocking, we note thatαplays an essential role in this model. This result brings out the importance of nonlinear processes in blocking events.

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Figure 10: Histograms resulting from the 1-D model with increasing nonlinear parameterα, with units of y-axis as relative density. Each event is LWA over a four day average. Horizontal lines denote the computed wave-breaking threshold for each value ofα(Uj= 60ms1−andγ= 3.0). (Whenα= 0.15 the wave breaking

threshold is greater than 70 and as such is not displayed.) Increasingαcorresponds with an increase in the tails of the histogram.

4 Two-Layer Quasigeostrophic Model

We will describe results from a two-layer model of baroclinic instability, which is based on quasigeostrophic dynamics like the one-dimensional model from the previous section is. This model was formulated by Norman Phillips (1951) — in his Ph.D. dissertation at the University of Chicago. We use the two-layer model to study the statistical properties of LWA in the storm track. A primary difference in the two models is that, unlike the one-dimensional model barotropic model considered earlier, the two-layer model spins up eddies through internal process of baroclinic instability. While the one-dimensional model is simple, eddy forcing in that model is prescribed. To best understand dynamics, ideally eddy forcing should emerge through an internal dynamics of the fluid without begin prescribed: for this, a two-layer model is required.

4.1 Model Setup

While similarly appreciated for its simplicity, the two coupled partial differential equations (PDEs) that form this model encompass all essential ingredients of the large-scale circulation of the troposphere in the midlatitudes: ∂q1 ∂t +J(ψ1, q1) = 0, q1=βy+∇ 2 Hψ1−L−D2(ψ1−ψ2), (4.1.1) ∂q2 ∂t +J(ψ2, q2) = 0, q2=βy+∇ 2 Hψ2+L−D2(ψ1−ψ2+ψB), (4.1.2)

whereqandψare potential vorticity (PV) and streamfunction, with subscripts 1 and 2 referring to the upper and lower layers, respectively. J is the 2D Jacobian operator,βy is the variation of the Coriolis parameter in latitude,∇2

H is the horizontal Laplacian,LDis the internal Rossby radius of deformation, andψB is the

perturbation to streamfunction due to bottom topography. In the linearized limit, Eqns. (4.1.1) and (4.1.2) permit analytical solutions for baroclinic instability [10].

Similar to Eady’s model, baroclinic instability is portrayed as arising from the interaction between two levels of the atmosphere [4]. Unlike Eady (and more similar to [3]), the Philips’s model incorporates the β−effect (latitudinal variation of the Coriolis parameter — and as such, latitudinal variation of planetary vorticity). Phillips subsequently extended this model for a long-term integration [?phillipsnum], considered by many as the first viable climate simulation. As previously mentioned, Philips’s model encompasses essential dynamical elements of the midlatitude circulation in the troposphere, yet its economy facilitates the isolation of key processes, which is more difficult in the use of more complex climate models.

We set up the model domain as a 28000 km ×28000 km rectangular channel, periodic in x(longitude) and bounded inyat north and south ends. We choosey= 0 to be the center latitude of the channel, i.e. the boundaries are at y =−14000km and y= 14000km. We assume thatβ = 1.6×10−11 _m−1_s−1 _and _L

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800 km. The model is solved with a spectral transform method, with nonlinear terms evaluated on the 256 ×256 grids in physical space. The governing equations are split into the zonal mean and eddy components to accommodate different boundary conditions at the rigid walls. The zonal-mean component, denoted by overbar, solves for the PV gradient and invert the zonal-mean zonal wind from it:

∂ ∂t _∂_q_¯ 1 ∂y =−∂ 2 ∂y2v 0 1q 0 1− 1 τRAD _u_¯ 1−u¯2−∆E L2 D +ν ∂ 6 ∂y6, _∂_q_¯ 1 ∂y , (4.1.3) ∂q¯1 ∂y =β− ∂2_u_¯ 1 ∂y2 + ¯ u1−u¯2 L2 D , (4.1.4) ∂ ∂t ∂q¯2 ∂y =−∂ 2 ∂y2v 0 2q 0 2+ 1 τRAD ¯ u1−u¯2−∆E L2 D + 1 τFRIC ∂2u¯2 ∂y2 +ν ∂6 ∂y6 ∂q¯2 ∂y , (4.1.5) ∂q¯2 ∂y =β− ∂2_u_¯ 2 ∂y2 − ¯ u1−u¯2 L2 D . (4.1.6)

In the above, the vertical shear ¯u1−u¯2 is relaxed toward the radiative equilibrium value

∆E= Λ sech2(y/3LD) (4.1.7)

with a radiative damping time ofτRAD = 30 days, whereas the bottom-layer wind ¯u2is damped by friction

with a timescaleτFRIC = 5 days. We also add 6th-order hyperviscosityν= 2.97×1024 m6s−1 to suppress

small-scale noise. The shear parameter Λ in (4.1.7) controls the strength of eddy (eddying increases with shear) and will be varied in the subsequent experiments. The model is initialized (apart from the zonal mean) with small-amplitude white noise with a meridionally symmetric structure.

The eddy components of the governing equations read: ∂q10 ∂t + ¯u1 ∂q10 ∂x + ∂ψ10 ∂x ∂q¯1 ∂y +J ψ1 0_{, q} 10−J ψ10, q10= 1 τRAD _ψ 10−ψ20 L2 D +ν∂ 6_q 10 ∂6_x , (4.1.8) q10=∇2Hψ10−L−D2 ψ10−ψ20 , (4.1.9) ∂q20 ∂t + ¯u2 ∂q20 ∂x + ∂ψ20 ∂x ∂q¯2 ∂y +J ψ2 0_{, q} 20 −J ψ20, q20 =− 1 τRAD _ψ 10−ψ20 L2 D − 1 τF RIC ∇2 Hψ20+ν ∂6_q 20 ∂6_x, (4.1.10) q20=∇2Hψ20+LD−2 ψ10−ψ20+ψB0 , (4.1.11)

A stationary wave is forced by a simple bottom topography with zonal wavenumber 2

ψB0(x, y) = Γ cos(4πx/L) sin(2πy/L), (4.1.12)

whereL= 28000 km. The height of the topography Γ will also be varied in the subsequent experiments. Also, the model is initialized (apart from the zonal mean) with a small-amplitude white noise in psi’ (same coefficient for all Fourier modes) with a meridionally symmetric structure.

4.2 Results

When understanding the results of the two-layer model of baroclinic instability, we are primarily interested in the distributions of LWA as well as the change in threshold value as we vary wind shear and topography. (The threshold value is equivalent to the value of LWA as discussed in the one-dimensional model, and can be considered the minimum LWA amplitude value for blocking behavior.)

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4.2.1 Distributions

Similar to observational data and results from the one-dimensional model, distributions of LWA are smooth, unimodal, with an evident right skew. In figure 11A, we show histograms LWA in the top layer of the model and vary wind shear as well as topography. The addition of topography (with Γ = 3×107_m2_s−1_{) changes}

the distributions fairly little, but slightly increases the height of the tails of the distribution, and also has an increased percentage of low values of LWA.

However, the distributions of LWA do change significantly as wind shear is increased. As wind shear increases from 10 m/s to 60 m/s, the high values of LWA become more common and extreme LWA values increase, essentially flattening the distributions. This result is somewhat expected, in that increased wind shear leads to increased eddying from baroclinic instability. Given that LWA is a metric of eddying, increased eddying is likely to lead to increased values of LWA.

Figure 11: Results from the Two-Layer Model of Baroclinic Instability. A. Histograms of top-layer LWA (at y = 0), with simulation run-length of 3 years, as wind shear is varied from 20 m/s to 60 m/s. The histograms for different types of topography are overlaid. B. Scatterplot of LWA vs. jet speed from the model where shear is 60m/s, overlaid with schematic depicting a sample calculation of the wave-breaking threshold as well as the zero wind value. C. Linear regressions of LWA vs. jet speed as wind shear is varied. Wave-breaking threshold and expected zero-wind values are marked. (Computed as in B.)

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4.2.2 Threshold Values

The scheme for computing threshold values of LWA for each simulation are shown in figure 11B. One can use a linear regression to compute the decreasing trend between LWA and jet speed (m/s, positive in the eastern direction), and then calculate the expected LWA value for zero wind (or the x-intercept of the linear fit). One half of the expected LWA is the wave-breaking threshold, and having zero (or negative) wind indicates a wind reversal, a marker of blocking.

We plot the linear regression, expected zero wind, and the wave breaking threshold in figure 11C, for model run with no topography. The results from simulations with topography showed the same relationship between increasing wind speeds and varied only slightly. The wave breaking threshold and the expected zero wind LWA value increased with wind shear. Thus, higher values of LWA do not necessarily imply more blocking behavior: although LWA is inversely correlated to transient jet speeds, higher average values of LWA do not necessarily imply lower values of jet speed or vice versa. In these simulations, jet speed also increased with wind shear (not an unexpected result).

Surpassing the wave breaking threshold is a prerequisite for blocking, and so if the distributions of LWA remained the same as wind shear varied, then an increased wind shear may lead to less blocking activity. However, given that the distributions flatten as shear increases, we see that increasing wind shear leads to more values of LWA above this threshold as well as above the zero wind marker. So if the jet speed weakens as a result of the reduced shear (north-south temperature gradient), it does not necessarily lead to more blocking events, as the eddy forcing is also reduced. This contrasts with the one-dimensional model in which we varied eddy forcing and jet speed independently; in the case of the (more realistic) two-layer model, eddy forcing and jet speed are not independent.

5 Discussion

LWA can measure the meandering of the jet stream, and persistent high values of LWA can denote atmo-spheric blocking. Blocking in the jet stream is anomalous weather events, it is not surprising that high values of LWA are linked to anomalies in both surface temperatures and geopotential height. As such understanding current distributions and what factors may affect values of LWA can give insight into extreme weather and how atmospheric dynamics in the midlatitudes may be affected if climatology changes.

For blocking to occur, values of LWA must pass the wave-breaking threshold. In the one-dimensional model, results indicate that sufficiently high amplitude of transient eddying as well as nonlinearity are essential for atmospheric blocking. Increasing the background jet speed did decrease LWA and prevent the wave-breaking threshold from being achieved.

On the other hand, in the two-layer quasi-geostrophic model, increases in wind shear (and also background jet speed) lead to both higher values of LWA as well as a higher observed wave breaking threshold. This is not necessarily a contradictory result: wind shear is essential to produce eddying from baroclinic instability. Eddying is a crucial component to blocking, and as such it is an unsurprising result that increased wind shear leads to the increased values of LWA. Further, it is promising that the negative relationship between LWA and wind speed in the two-layer model are similar to the covariance between wind speeds and LWA in observational values as well as in the one-dimensional model result.

From these results one may speculate on the effects of climate change on the behavior of the jet stream. If the average wind shear of the jet stream is to decrease (a highly debated possibility in the case of rising temperatures), we can hypothesize that increased eddying from baroclinic instability could decrease LWA values and blocking events. However, given that mean LWA may also decrease, it could decrease blocking behaviors. The competition between the two is one of the reasons why the future projection of blocking frequency by climate models is very much inconclusive.

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What are the Controls on Distributions of Local Wave Activity?